where \(E_{\rm LV}\)  is the Lorentz violation scale, \(K\)  is the LV factor
and \(s=\pm 1\)  is the sign factor of the LV correction term. Since the linear LV correction implies the \(CPT\) -odd term in an effective field theory, neutrinos and anti-neutrinos have opposite sign factors, i.e.,
there is an asymmetry between neutrinos and anti-neutrinos [1]}, [2]}.
This time criterion requires that the observed time differences between the IceCube event and its associated GRB satisfy the possible Lorentz violation regularity
proposed from GRB photons, TeV and PeV neutrinos, with the consideration of errors.
The maximum correlation criterion and the strict time criterion can effectively
depress the effects of GRB backgrounds. While considering the backgrounds, one should
focus on the refined strict time criterion, rather than a rough time window of
months.
For the directional criterion, a two dimensional circular Gaussian [3]}
\(P(\nu , \mathrm {GRB})=\frac{1}{2\pi \sigma ^2}\exp (-\frac{\Delta \Psi ^2}{2\sigma ^2}),\) 
